Physics – Data Analysis – Statistics and Probability
Scientific paper
2006-05-10
Phys. Rev. E 74, 036104 (2006)
Physics
Data Analysis, Statistics and Probability
22 pages, 8 figures, minor corrections in this version
Scientific paper
10.1103/PhysRevE.74.036104
We consider the problem of detecting communities or modules in networks, groups of vertices with a higher-than-average density of edges connecting them. Previous work indicates that a robust approach to this problem is the maximization of the benefit function known as "modularity" over possible divisions of a network. Here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix, which plays a role in community detection similar to that played by the graph Laplacian in graph partitioning calculations. This result leads us to a number of possible algorithms for detecting community structure, as well as several other results, including a spectral measure of bipartite structure in networks and a new centrality measure that identifies those vertices that occupy central positions within the communities to which they belong. The algorithms and measures proposed are illustrated with applications to a variety of real-world complex networks.
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